Kolmogorov's Heritage in Mathematics

• The youth of Andrei Nikolaevich and Fourier series.- Kolmogorov’s contribution to intuitionistic logic.- Some aspects of the probabilistic work.- Infinite dimensional Kolmogorov equations.- From Kolmogorov’s theorem on empirical distribution to number theory.- Kolmogorov’s ?-entropy and the problem of statistical estimation.- Kolmogorov and topology.- Geometry and approximation theory in A. N. Kolmogorov’s works.- Kolmogorov and population dynamics.- Resonances and small divisors.- The KAM Theorem.- From Kolmogorov’s Work on entropy of dynamical systems to Non-uniformly hyperbolic dynamics.- From Hilbert’s 13th Problem to the theory of neural networks: constructive aspects of Kolmogorov’s Superposition Theorem.- Kolmogorov Complexity.- Algorithmic Chaos and the Incompressibility Method.

From the reviews: "Andrei N. Kolmogorov (1903-1987) was one of the most prolific and influential mathematicians of the twentieth century. ... this collection would be read by those who hold only a bachelor's degree in mathematics, computer science or physics, and more generally, by those interested in mathematical ideas. ... This impressive collection gives the reader an understanding of the depth and breadth of Kolmogorov's contributions to mathematics. Each chapter includes a substantial list of references that invite further reading." (Elena Anne Marchisotto, MathDL, December, 2007) "The object of Kolmogorov's Heritage in Mathematics is to enable the general mathematical reader (educated to master's level) to grasp something of Kolmogorov's achievements. ... I can judge, it is successful in its aim. ... those mathematicians with a sense of history, who wish to place themselves within the general culture of mathematics, will find this book a worthy memorial to a great man." (T. W. Korner, SIAM reviews, Vol. 50 (4), December, 2008)